The Collatz conjecture, which states that repeated application of the function ( f(n) = begin{cases} n/2, & text{if } n equiv 0 pmod{2} 3n+1, & text{if } n equiv 1 pmod{2} end{cases} ) to any positive integer ( n ) will eventually reach the number 1, has been a long-standing open problem in mathematics. In this paper, we investigate a generalized version of the Collatz function, denoted as ( f(E, T) ), where ( E ) is a positive integer and ( T ) is a fixed positive integer. We prove that for any positive integer ( E ), repeated application of ( f(E, T) ) will eventually lead to an even number. Furthermore, we show that any even number will eventually reach a power of 2 under repeated application of ( f(E, T) ), and once a power of 2 is reached, the sequence will enter the cycle ( 1 ightarrow 4 ightarrow 2 ightarrow 1 ). These results provide new insights into the behavior of the generalized Collatz function and its convergence properties.